New coupled equations describing collisions of an atom and a diatomic molecule are derived in this paper. By utilizing a description of the collision in terms of rotating coordinates, all coupling in the z component of angular momentum is isolated into purely kinematic effects. By neglecting these couplings, one is led to approximate equations for which the jz component of angular momentum for the molecule is conserved. In addition, the scattering cross sections are formulated by neglecting the effect on the wavefunction of the rotation of the coordinate axes so that in place of Wigner rotation matrices dmmJ (Θ) appearing, one deals with simple Legendre polynomials and the orbital angular momentum l2 is approximated by l(l + 1) ℏ2. It is noted that the procedure involves no approximations so far as the potential matrix elements are concerned. Furthermore, the number of equations remaining coupled is drastically reduced and a completely quantum mechanical description of the dynamics of both internal states and relative motion is retained. The physical implications of the approximations are examined, and it is seen that the neglect of intermultiplet coupling gives rise to consideration of only transitions where both the orientation and magnitude of the rotor angular momentum change. Further, the neglect of transformation effects on the wavefunction is expected to be least accurate for the inelastic forward scattering and best for backward scattering and the j =0→0 elastic scattering. Finally, the present simplest version of the approximation obviously is not intended for treating processes dependent on mj transitions, e.g., NMR relaxation in He–H2. Next the formalism is applied in test calculations to He–H2 collisions using the Krauss-Mies potential energy surface. Numerical results for elastic and inelastic integral and differential cross sections are compared with exact quantum mechanical close coupling solutions of the standard coupled channel equations. Over the energy range studied (from 0.1 eV up to 0.9 eV), agreement to within a few percent is obtained. Additional coupled states calculations are reported at 1.2 eV and computation times are compared against those required for a full close coupling solution. Calculations for the Roberts He–H2 surface are also reported to illustrate the independence of the approximations on the strength of the coupling (so long as the inelastic scattering is predominantly in the backward direction). The dramatic savings afforded by the present approach are such as to make possible fully converged calculations at collision energies typically studied in molecular beam experiments. Thus, for elastic and inelastic nonreactive collisions, involving a repulsive-type interaction, the approach makes the a priori quantum mechanical description of the scattering of a diatom by an atom practical.

disagreement between the vibrational levels of it and those of the Bowman potential and for remaining uncertainties in the assignment of the observed levels. We have suggested in the above analysis a corrected well depth of >1075 cm'1 11for the Bowman potential. This is to be compared with the ab initio well depth of -1109 cm'1. There is clearly reasonable agreement among these two values and the 1062-cm"1 well depth of the Bowman potential. There is, however, a large discrepancy between the zero-point levels of the two surfaces that arises primarily from what appears to be an overly large contribution (~440 cm"1) from the bending vibration on the ab initio surface. This leads not only to a systematic shift in the computed vdW stretching levels compared to experiment, but to the prediction of fewer excited bending states.The fact that experimental estimates of the dissociation energy, which range from 718 to 742 cm"1, are so closely in accord with the present calculation (724 cm"1) and estimation (~739 cm"1) on the Bowman surface strongly suggests that the bending po-tential is better described by that surface. This inference is independent of the manner in which the non-vdW stretching levels are assigned. The well depth of the ab initio surface would have to be in error by enough to allow a 126-cm'1 increase in the dissociation energy. While Chakravarty and Clary10 are inclined to accept a 15% error in the ab initio well depth, one must expect such an error also to affect the bending potential.

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrodinger equation. Since the ground state of each partner potential is nodeless, we avoid any "node" problem typically associated with the Monte Carlo technique. Although we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.

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